G99/3

The Forward Rate Unbiasedness HypothesisIn Inflation-Targeting Regimes

W A Razzak1

First Version November 1998Revised August 1999

F3, F4, C1

Abstract

I test the forward rate unbiasedness hypothesis using the Error Correction Model (ECM)of Naka and Whitney (1995). It is shown that the Naka-Whitney treatment of the dynamicis perhaps necessary to ameliorate existing problems associated with testing thehypothesis. However, it is not sufficient due to the sensitivity of the non-linear regressionto starting values. Monthly data from October 1985 to May 1998 for New Zealand,Canada, United Kingdom (UK), Sweden, Germany, Japan and South Africa are used totest the forward rate unbiasedness hypothesis. The first four countries are inflationtargeting regimes. Germany and Japan have both had very low inflation and South Africaexperienced periods of very high inflation followed by periods of low inflation. The nullhypothesis is widely rejected. The premia puzzle remains largely unexplained.Interestingly, the forward rate unbiasedness hypothesis holds in two inflation-targetingregimes namely New Zealand and UK. Implications of the Consumption – Asset PricingModel (CAPM) are used to explain the finding.

1 I am very thankful to Christopher Smith for his contribution to this research. The paper also benefited

from comments by Yin-Wong Cheung, Francisco Nadal De Simone, Clive Thorp, Andres Vredin, JohnMcDermott and David Giles. Also, I would like to thank Arthur Grimes, Young-Sik Kim, Bob Buckle,Nelson Mark, Pierre Siklos, Brend Hayo, and Stephen Millard. Comments made in the seminar atAntwerp University-Belgium (1998), the workshop on monetary policy uncertainty held at the CentralBank of Finland in September 1998 and the Far East Meeting of the Econometric Society in Singapore1999 were very helpful. The views expressed in this paper do not necessarily represent those of theReserve Bank of New Zealand.

1 Motivation, theory, and empirical issues

The primary objective of this paper is to examine the forward rate unbiasednesshypothesis in New Zealand under the current regime of inflation targeting. I also examinethe hypothesis in other inflation-targeting regimes (e.g., Canada, UK and Sweden) andcompare the results to some non inflation-targeting countries (e.g., Germany, Japan andSouth Africa).

Several reasons motivated this research. One is that Naka and Whitney (1995) seem tohave resolved the forward premium’s anomaly in the G7 countries. Thus, provided theencouragement to test it for New Zealand, particularly since the hypothesis does not seemto have been tested using recent New Zealand data. Second, a reasonably long time seriescovering the period of inflation targeting became available, which made time seriesanalysis feasible. Third, ever since New Zealand floated its currency in March 1985, theReserve Bank of New Zealand has never actually intervened in the exchange market,which might lend support to the efficiency argument. Fourth, in 1989 New Zealandofficially adopted an inflation-targeting regime, which altered the time series properties ofmacroeconomic data, the moments and cross moments (e.g., inflation became stationaryand the covariance between inflation and the depreciation rate became zero). It will beshown that these changes can significantly affect the relationship between the forward rateand the expected spot rate.

For comparison, some other inflation-targeting regimes are added to the sample. Thehypothesis is also tested for the Canadian dollar (CAD/USD), the Pound (GBP/USD), andthe Swedish Krona (SDK/USD).2 Data form Germany (DM/USD) and Japan (YEN/USD)are also tested. Although these two countries have no institutional arrangement forinflation targeting, they have experienced periods of very low-and-stable inflation in the1980s and the 1990s. The hypothesis is also tested for the South African Rand(SAR/USD). South Africa experienced high inflation in the 1980s and low inflation inthe 1990s, but including it in the sample serves as a control variable. The hypothesis isalso tested for all currencies with respect to the Deutsche Mark (DM).

To outline the theory behind the forward rate unbiasedness hypothesis I will start withsome notations. Let et be the natural logarithm of the spot exchange rate, where the level

of the spot rate,tε is defined as the domestic currency/US dollar (USD). Similarly, let

f t be the natural logarithm of the one-period forward exchange rate, tF . The domestic

currency annualised 30-day interest rate is ti and the foreign interest rate is*ti . The price

level tP is defined by the natural logarithm of the CPI. The covered interest rate arbitrage

at one-horizon ensures that:

*tttt iief −=− (1)

2 Sufficient time series for other inflation-targeting countries such as Spain, Finland and Israel are not

readily available.

Equation (1) must hold since an investor at home can borrow one unit of the domesticcurrency by buying − et worth of USD, make an investment in US-denominated bonds

paying *ti per bond and then secure a future domestic currency unit pay-out by selling the

return forward at a rate f t .3 In general, and particularly in a world without capital control,such a transaction bears no or little risk because it can be carried out at time t . Therefore,gross payments and costs of the initial dollar borrowing must equalise.

The question of interest is whether or not the following relationship

)|( 1 ttt eEf Ω= + (2)

holds, where E is the rational expectations operator given an information set Ω t . If (2)does not hold, investors can earn arbitrarily large profits by speculating in forward foreignexchange markets and the arbitrage theory is violated.

If the rational economic agent expects the spot exchange rate in the future to be the sameas the current forward rate then she will be indifferent between the two transactions undercertain conditions. These conditions are (1) that agents are risk-neutral, and (2) that thereare identical transaction costs in forward and spot markets. It is therefore important todistinguish between the two hypotheses, rational expectations and the unbiasedness of theforward rate.4

The unbiasedness hypothesis has been tested by estimating the following equation:

e a b f ut t t= + +−1 . (3)

3 There are issues related to what is called “Siegel’s Paradox,” but they are not important for this paper.

See Obstfeld and Rogoff (1999, p. 586) for details.

4 The first is referred to as the “efficient market” hypothesis. This says that economic agents use allavailable information to forecast the future spot rate, i.e., they form rational expectations of the futureexchange rate. The market efficiency hypothesis states that the market behaves as if traders possessedrational expectations. Market efficiency implies:

,11 ++ += tttt eEe φWhere 1+tφ is a serially independent forecast error with mean zero. Moreover, the error term is

uncorrelated with any observed lagged variable. This hypothesis is not directly testable because of theexpectations term on the right-hand-side, which is unobservable. Bilson (1981) introduced the secondhypothesis that is the “speculative efficiency hypothesis,” which is basically equation (2). The validityof this equation depends on the two conditions stated earlier, risk-neutrality and identical transactioncosts in forward and spot markets. Combining equations (2) and the above equation together give:

ttt zfe += −1 ,

where tz is an error term. Tests of this equation are not tests of “market efficiency” or “rational

expectations” because this equation could only be derived with the auxiliary assumptions of risk-neutrality and identical transaction costs. There are two hypotheses in the above equation: rationalexpectations and the equality of the forward and the expected spot rates. Nevertheless, a form of thisequation has been used to test for the unbiasedness hypothesis.

The parameter a is interpreted as a constant risk premium. To test the hypothesis that theforward rate is an unbiased predictor of the spot rate, the restriction 1=b is tested.Various estimation techniques have been used to test this hypothesis. A stronger form -unbiasedness-market efficiency hypothesis and no risk premium - implies testing 0=a ,

1=b and that the ut ’s are serially uncorrelated and homoscedastic.5 Wolf (1987) shows

that if the restriction 1=b holds, serial correlation in the tu ’s is still consistent with a

time-varying risk premium. Note that when equation (3) is estimated by OLS and when ut

is serially uncorrelated according to the DW statistic, the magnitude of b is usually foundto be equal to one.

Developments in the literature on unit roots-cointegration suggest that researchers shouldnot estimate equation (3) by OLS because of the presence of unit roots in the spot and theforward rates, which render standard test statistics not useful.6 The OLS parameters ofequation (3), though, are super-consistent. Although it seems reasonable to use a FullyModified OLS (FM-OLS) estimation technique, but Phillips and McFarland (1997) argueit may not be appropriate to estimate equation (3). Provided that the spot rate and theforward rate are cointegrated, the following ECM has been used instead:7

e e a b f e ut t t t t− = + − +− − −1 1 1* * *( ) . (4)

Again, testing the unbiasedness-market efficiency hypothesis involves testing 1* =b ,( a b* *;= =0 1) jointly, and thatut

* is serially uncorrelated and homoscedastic. Hodrick(1987), Lewis (1995), and Engel (1996) document the empirical failure of equation (4) insurveys. Most researchers find b* to be negative. Froot and Thaler (1990) report that theaverage estimate of b* across 75 published estimates is -0.88. The fact that the spot andthe forward rates are cointegrated, that b in equation (3) is 1, and yet *b in equation (4) isnegative presents a problem. These empirical findings cannot be reconciled witheconometric theory.

Fama (1984) offers an explanation of why the OLS coefficient b* < 0 . He argues that therational expectations risk premium on foreign exchange rates must be extremely variable.Also, a large number of empirical papers suggest that conditional variances of exchangerates vary over time.8

Cumby and Obstfeld (1981) assume that the risk premium separates the forward rate fromthe market expected future spot rate. Hansen and Hodrick (1983) use a CAPM type

5 For example, see Frenkel (1977), Edwards (1983), and Baillie et al (1983).

6 For example, see Messe and Singleton (1982) and Baillie and Bollerslev (1989).

7 For example, see Baillie and Bollerslev (1989), Hakkio and Rush (1989), Barnhart and Szakmary(1991), Liu and Maddala (1992), and Naka and Whitney (1995).

8 For example, see Cumby and Obstfeld (1984), Hsieh (1984), Domowitz and Hakkio (1985), Mark(1985, 1987), and Lyons (1988).

model to test the same hypothesis. These models imply that the forward rate is not anunbiased predictor of the spot rate. The wedge between the forward rate and the expectedspot rate consists of the variance of the spot rate, the covariance between the spot rate andthe price level, and the covariance between the spot rate and real consumption. They findthat the risk premium is only important in two out of the five exchange rate markets theytest. Frankel (1980) investigates the risk premium in six currencies and fails to find it.Frankel (1986) argues that the risk premium must be very small in magnitudes. So far,this model has had limited success in explaining the anomaly of the forward premium.

McCallum (1994) provides a different explanation. He explains the failure of the forwardrate unbiasedness hypothesis by a neglect to take into account the fact that monetaryauthorities pursue interest rate smoothing policies, and avoid exchange rate changes.Therefore, he suggests model (3) and (4) have a missing equation that accounts for thebehaviour of the monetary authority. Thus, the models are misspecified.

Naka and Whitney (1995) argue that the dynamics in the error correction model inequation (4) are misspecified. They argue that it is this misspecification that causesestimates of b* to be negative. They derive another error correction model directly fromequation (3). The ECM is derived as follows.

From equation (3) we get

u e b f at t t− − −= − −1 1 2 . (5)

The only additional assumption is that the error term tu is given by tt vu +−1ρ . By

substituting this into equation (5), they obtain:

e b f a e b f vt t t t t− = − + − +− − −1 1 21( ) ( )ρ ρ . (6)

The unit root in the forward rate implies:

f ft t t− − −= +1 2 1η . (7)

Subtracting et −1 from both sides of equation (6), they arrive at a new dynamicspecification:

ttttttt vffbefbaee +−+−−+−=− −−−−− )())(1()1( 21121 ρρ (8)

Note that if the error terms vt and ηt −1 are (i.i.d.) and uncorrelated then equation (6) and(7) form a triangular system of the type introduced by Phillips (1991) and Phillips and

Loretan (1991).9 The Naka and Whitney equation (8) should not be viewed as a purelystatistical model; similar dynamic equations are observed in CAPM.

Equation (8) seems to ameliorate the existing problems in equation (4) and explains theempirical anomaly rather well. The hypothesis seems to hold in Naka and Whitney’ssample of the G7 countries from 1974 to 1991. However, it will be shown in this paperthat this procedure is necessary but not sufficient for the null hypothesis to hold. Thus,the forward premium puzzle is still largely unexplained. Interestingly, the hypothesisseems to hold in inflation-targeting regimes, which needs some explanation.

The next section of the paper presents the estimation results using equation (4) and theNaka-Whitney approach (equation (8)). Results suggest that the hypothesis may hold wellin many countries and over different samples. Not surprisingly, non-linear estimationtechniques based on numerical evaluations algorithm are sensitive to starting values. Itwill be shown that when the global optimum is distant from the hypothesised values( 0=a , 1=b and 0=ρ ), numerical procedures will typically converge to a localoptimum rather than – the more distant – global optimum. It is shown that the resultsfound by Naka and Whitney are not robust and the forward rate unbiasedness hypothesisis rejected in many cases and across different sub-samples. Interestingly, the hypothesiscould not be rejected for some inflation-targeting regimes such as New Zealand, UK and,probably, Sweden. Section 3 offers an explanation for the findings using an implicationof the CAPM. Section 4 contains a summary.

2. The data and testing the unbiasedness hypothesis

It is important to note that he sample used in this paper is longer than that used in Nakaand Whitney (1995) - monthly data from January 1985 to May 1998. Their sample isfrom January 1974 to April 1991 so their sub-sample from October 1985 to April 1991overlaps with my sample. Some non-trivial changes in money, interest rates, and inflationwere associated with changes in Europe since 1991 that have direct relevance to theforward rate unbiasedness hypothesis. The spot rates (end-of-period), and the forwardrates (end-of-period) are from Datastream (Barclays’). The forward rates are 30-dayforward contracts. Seven exchange rates are used. Four are inflation-targeting countriesnamely New Zealand, Canada, UK and Sweden. Two are low-inflation countries,Germany and Japan. Finally, I added South Africa, which experienced periods of veryinflation in the 1980s and low inflation in the 1990s. These currencies are the NZD/USD,CAD/USD, GBP/USD, SDK/USD, DM/USD, YEN/USD and SAR/USD. The exchangerate in each case is defined as the US dollar price of one unit of each of the currencies.

9 v

i i dt

tη −

≈

1

0. . .( , )Σ , Where Σ =

2

2

0

0

ησσ v

and if tv fand,0, =ησ is strictly exogenous (Naka

and Whitney (1995)).

Also, the same hypothesis is tested using the exchange rates defined as the DM price ofone unit of each of the currencies.

Baillie and Bollerslev (1989), Hakkio and Rush (1989), Barnhart and Szakmary (1991),Liu and Maddala (1992) and Naka and Whitney (1995) provide evidence that the naturallogarithms of the spot exchange rate and the forward rate are unit roots and arecointegrated. Similar results are found in this data set. Results are not reported to savespace, however, they are available upon request.

The next few pages contain the parameter estimates and the analysis of the results.Results are reported in tables 1-6. The first three tables are related to equation (4) and (8).The findings are summarised as follows. There is a clear difference between theparameter estimates of equation (4) and (8). Equation (4) results in the usual finding that

*b is negative. However, the parameterb in equation (8) is no longer negative, but closerto unity in magnitude, which confirms the Naka and Whitney (1995) argument that thefailure of the unbiasedness hypothesis was due to the misspecified dynamic of equation(4). However, unlike the results reported in Naka and Whitney (1995) these resultssuggest that the hypothesis hold for samples covering the 1990s, but not during the 1980s.It is important to remember that the full sample in this paper is longer than that of Naka-Whitney. The regressions for the 1990s indicate no significant ARCH effects, but those of1980s show some ARCH effects. The parameters are clearly unstable, as the Chow testsindicate.10

Equation (4) – the traditional ECM – is first estimated using OLS. The negative resultsreported in the literature are confirmed. The results are reported in table 1. The P-valuesfor the Wald statistics are reported for all hypotheses. For the full sample, the hypothesisthat a* = 0 cannot be rejected except for the YEN/USD. The hypothesis that b* = 1 isrejected in all cases except in the case of SDK/USD (i.e., b* =1.54) and DM/USD (i.e.,b* =1.04). The joint hypothesis that 0* =a and 1* =b , is rejected by the data except inthe cases of SDK/USD and DM/USD.

Second, for each inflation-targeting country, the sample is split into an inflation-targetingperiod and a pre-targeting period, and the same hypotheses are tested for each sub-sample.11 For the other countries the break points of Naka and Whitney (1995) are used.The Naka-Whitney first sub-sample covers the 1980s and is from October 1985 to April1991. The second sub-sample covers the 1990s and is from May 1991 to May 1998. Forthe pre- inflation targeting sub-samples during the 1980s, the hypothesis that a* = 0

10 The Chow test may not be applicable in the presence of ARCH.

11 The inflation-targeting period for New Zealand is taken from January 1989, which is the date when thePrice Targeting Agreement was signed, to 1988. It has been argued that the Reserve Bank of NewZealand started targeting lower inflation at least two years earlier. For Canada, I use January 1991 as astarting date. Inflation targeting in the UK started in October 1992. Inflation targeting starts January1993 in Sweden. Note that both the UK and Sweden pegged their currencies to the DM prior toinflation targeting, up until September 1992 and November 1992 for the UK and Sweden respectively.

cannot be rejected except in the case of Sweden. The null hypothesis that *b is equal toone is also rejected by the data.12

Results for the inflation targeting periods of the 1990s are slightly different from theperiods of pre inflation targeting and the 1980s. The UK left the Exchange RateMechanism (ERM) in September 1992. This seems to introduce some volatility in thedata so the estimation starts in October 1993, i.e., a year later. Also, note that Swedenfloated its currency in November 1992 so estimation starts from December 1993. Thehypothesis that *a is equal to zero cannot be rejected for any country.

Interestingly, the hypothesis that *b is equal to one cannot be rejected for New Zealandand the UK. The joint hypothesis that 0* =a and 1* =b also cannot be rejected for NewZealand and the UK. To emphasis that the results for the UK and New Zealand aredifferent from all other currencies, and that they are obtained from the traditional ECM, Ireported them here. The numbers in parentheses are the P-values of the Wald statistics.

*11

**1 )( ttttt uefbaee +−+=− −−−

Period NOB *a *b )1,0(:0 ** == baH DW

NZD/USD 89:1-98:5 113 0.003(0.31)

1.46(0.53)

(0.58) 1.71

GBP/USD 93:10-98:5 56 0.002(0.46)

0.61(0.87)

(0.54) 2.22

Next, equation (8) – Naka and Whitney’s equation – is estimated using NLLS for the fullsample and over the sub-samples defined above. The starting values for the parametersa , b , and ρ are zero, one and zero respectively. The results are reported in table 2.13

There are marked differences between the magnitudes of the b* ’s in equation (4) and theb ’s in equation (8). The b ’s are closer to unity in magnitudes while the b* ’s in equation(4) were either negative or greater then unity in magnitude. To a large extent, thisconfirms Naka and Whitney (1995) thesis.

In the full sample, the Wald statistic P-values indicate that a ≠ 0 in all countries exceptCanada and South Africa. Although the estimated b ’s are reasonably close to unity inmagnitudes, they are significantly different from unity except for Canada and SouthAfrica. Further, the joint hypothesis that a = 0 and b = 1 does not hold, again except forCanada and South Africa. The hypothesis that ρ = 0 cannot be rejected and the DWstatistics’ P-values, indicate that the residuals are serially uncorrelated. Also note that

12 Interpretation of the results should be affected by the presence of ARCH in the first sub-sample.

13 Using the ML method to estimate the regression does not seem to change the results. This confirmsNaka-Whitney.

New Zealand, the UK and Sweden have statistically significant Auto-RegressiveConditional Heteroskedasticity.

Similar results are found in the sub-sample covering the 1980s. The P-Values of theWald statistic indicates that the hypothesis a = 0 is rejected in all countries except NewZealand, Canada and South Africa. The slope parameter b is close to unity in magnitude,but statistically different from unity for all countries except New Zealand, Canada andSouth Africa. The joint hypothesis that a = 0 and b = 1 is rejected for all countriesexcept South Africa. The hypothesis that ρ = 0 cannot be rejected and the DW P-valuesindicate that the residuals are serially uncorrelated. The ARCH effects have disappearedfrom all regressions, except in Sweden and South Africa.

Note how the results change during the 1990s and for all countries. In the second sub-sample, the Wald statistic P-values indicate that the null hypothesis that a = 0 cannot berejected in all cases. The null hypothesis that b is equal to one cannot be rejected.Again, the joint hypothesis that a = 0 and b = 1 cannot be rejected in all cases. Also, theresiduals are serially uncorrelated except, may be, in Japan. The ARCH effectsdisappeared completely. These results indicate that there is much more support for theforward rate unbiasedness hypothesis during the 1990s rather than the 1980s. Chow testsindicate significant instability in all cases except for the DM/USD exchange rate (Chowtest results are not reported, but they are available upon requests).

The same exercise is repeated using the exchange rates with respect to the DM instead ofthe USD, with the results reported in table 3. Qualitatively, the results are the same asthose presented in table 2. The Chow tests indicate parameter instability in the cases ofNZD/DM and the GBP/DM only.

In summary, there is a clear difference between the parameter estimates of equation (4)and (8). Equation (4) results in the usual finding that *b is negative. However, theparameter b in equation (8) is no longer negative, but closer to unity in magnitude, whichconfirms Naka and Whitney’s (1995) argument that the failure of the unbiasednesshypothesis was the result of the misspecified dynamic in equation (4). Unlike theevidence presented in Naka and Whitney (1995), where the hypothesis held acrossdifferent sub-samples, in this paper the hypothesis seems to hold during the samplescovering the 1990s, but not during the 1980s. The regressions for the 1990s indicate nosignificant ARCH effects, but those of 1980s show some ARCH effects. Parameterinstability is present as indicated by the Chow tests.Naka and Whitney (1995) suggest that their results are robust to various estimationtechniques such as Maximum Likelihood and Non-linear least squares. But the fact thatequation (8) is a non-linear regression make it sensitive to starting values. Next, thissensitivity of the regression to starting values and small sample is checked.

The sum of squared residuals (SSR) in equation (8) is non-linear function of theparameters. In the absence of analytical solutions to the first order conditions of thisproblem, numerical methods must be used to minimise the residual sum of squares.However, the commonly used algorithms are subject to starting point problems. In the

presence of multiple possible values one cannot guarantee that such algorithm willconverge to the global optimum. Suppose that SSR is a function of ba,( and )ρ , and

suppose that 11,( ba and )1ρ is the argmin of this function. The standard numericalmethods find the parameter vector that sets the gradient of the function equal to zero.However, if 22 ,( ba∃ and )2ρ that sets the gradient equal to zero, and if the this parametervector is close to the starting values used to initialise the algorithm, then the globalargmin will not necessarily be recovered.

There is however, a robust solution for this problem. For a known value of ρ , it can beshown that the regression becomes linear of the following type as in Davidson andMacKinnon (1993, chapter 10).

ttt

tt ffbaee ω

ρρ

ρ +

−

−=−+∆

−−−

211

1],[)1( , (9)

Under such circumstances the OLS parameter estimates minimise the sum of squareresiduals. Thus, a grid search over the values of ρ is performed and the parameters a andb that minimise the sum of square residuals are estimated. Also, Wald tests and their P-values can also be obtained.14 Because all of the samples used in this paper are small,particularly the sub-samples, a bootstrap procedure is used to develop the distributions ofthe various Wald statistics.15

Equation (9) is estimated. The results of the non-linear regressions based on the Naka andWhitney approach that I reported in tables 2 and 3 have changed. New results are

14 Although, in the non-linear context, the distribution of the Wald statistic is only asymptotically 2

κχ(κ independent restrictions) the residuals are divided by 3−T to try to be relatively conservative

about the magnitude of the standard errors. Dividing by 3−T will be equivalent asymptotically to

dividing by T . If restrictions are only applied to a subset of parameters the variance-covariancematrix of the subset of the parameters is used. The variance-covariance matrix is estimated by thedelta method with partial derivatives being evaluated at the “true” hypothesised parameter values.

15 Given the joint hypothesis that 0=a and 1=b , a vector of standard errors is used as the populationfrom which Pseudo random errors are drawn. Pseudo data are formed by randomly sampling (withreplacement) from this population of errors and adding them to the lagged forward rates. In thisexercise the forward rates are assumed exogenous. The Pseudo spot rates at time t are formed bytaking the previous period’s forward rates and adding to them stochastic error terms, which wereobtained by sampling at random from the population of estimated errors. Thus, the r th data sample isformed by implementing the following:

rtt

rt vfe += −1 ,

where rtv is sampled at random. The equation is implemented for Tt ,...2= , where the first

observation is set equal to the first observed spot rate; each Pseudo sample thus has the same firstobservation. From this Pseudo data, and the forward rates that are held constant across all Pseudo-samples, one can estimate the above equation using the grid-search procedure, obtaining Pseudoparameters and the corresponding Wald statistics. Five thousand Pseudo data samples are replicatedand from this exercise one can generate empirical distributions for the parameters and the Waldstatistics.

reported in tables 4 and 5. It is found that the joint hypothesis 0=a and 1=b is rejectedin all cases except for the SDK/USD with a Wald P-value equal to 0.083 in the fullsample. The hypothesis cannot be rejected for the DM/USD in the first sub-sampleduring the 1980s, but only marginally so with a Wald P-value equal to 0.068. Germanyexperienced lower inflation rates in the 1980s than in the 1990s.

Two interesting cases are found. First, the hypotheses that 0=a and 1=b cannot berejected in the case of NZD/USD during the period of inflation targeting with Wald P-values of 0.282 and 0.191 respectively. The joint hypothesis that 0=a and 1=b cannotbe rejected either with a Wald P-value equal to 0.433. Thus, the evidence in favour of theunbiasedness hypothesis is significant. The regression reaches a global optimum with avalue of ρ equal to 0.13. Second the hypotheses that 0=a and 1=b cannot be rejectedin the GBP/USD cases during the period of inflation targeting with Wald P-values 0.053and 0.071 respectively. The joint hypothesis that 0=a 1and 1=b cannot be rejectedeither with a Wald P-value equal to 0.256. Again, the unbiasedness hypothesis holdswell. The regression reaches a global optimum with a value of ρ equal to -0.05. Recallthat the NZD/USD and GBP/USD performed equally well in equation (4) (see tables 1).

The hypotheses are rejected in all cases where the exchange rates are defined in terms ofthe DM except for YEN/DM during the 1990s. Results are reported in table 5.

Figures 1 through 7 show the difference between the local and the global optimum for allthe cases involving the USD during the second sub-sample in the 1990s. The hypothesesare rejected when the local optimum – rather than the global optimum – is in the vicinityof ρ equal to zero. The sums of square residuals (SSR) are plotted against grid of ρvalues for all seven cases covering the second sub-sample in table 4.

To summarise, the evidence that the forward rate is an unbiased predictor of the expectedspot rate is clear in the NZD/USD and the GBP/USD over the inflation targeting sub-samples. This is evident in the results reported in the old ECM in table 1 and in the newECM reported in table 4. Weaker, but still significant evidence in favour of theunbiasedness hypothesis is found in the GBP/USD during the inflation-targeting period.Also, the hypothesis seems to hold in the SDK/USD in the first sub-sample and the fullsample. The evidence in favour of the hypothesis is marginally significant in theDM/USD during the period of low inflation in Germany during the 1980s. The moreelaborate dynamic specified in the ECM of Naka and Whitney (1995) is perhaps necessaryto ameliorate existing problems associated with testing the hypothesis. However, it is notsufficient. The favourable results found by Naka and Whitney (1995) are largely due tothe numerical techniques used to estimate their ECM.

3 Implications of CAPM

Can we explain these results? Baxter (1994, p. 18) emphasises that short-run movementsin the exchange rate are likely to be determined by monetary factors (e.g., inflation).Bernanke and Mishkin (1997, p. 106) argue that uncertain inflation exacerbates relative

price volatility (the exchange rate is a relative price). So, a higher variation in inflationmight be associated with a higher variation in relative prices (i.e., the exchange rate).

The Consumption – Asset Pricing model (CAPM) may provide an answer because it linksthe forward rate unbiasedness to the variance of the spot rate (see Obstfeld and Rogoff,1996, p 585-592). They show that under the assumption that the forward rate tf , the spot

rate te , real consumption tC , and the price level tP are jointly lognormally distributed, the

following relationship can be derived:

),(),()(2

1)( 111111 ++++++ −−=− ttttttttttt CeCovPeCoveVareEf θ , (10)

where tE is the rational expectations’ operator based on the information set at time t andθis the constant relative risk-aversion elasticity.

Equation (10) holds for a risk-neutral investor if θ =0. The last term on the RHS is thetrue risk premium. To calculate the terms on the RHS of equation (10) the first differenceof the data is used because of the unit roots.

For the forward rate to be equal to the expected spot rate, the RHS terms should be eitheridentically zero or very close to zero. Some insight can be gained from the informationreported in table 6.

First, it is widely accepted that the risk premium (the third term in equation (10)) is verysmall in magnitude. This is because changes in consumption are not usually veryvariable, while changes in the spot rate are (see Obstfeld and Rogoff, 1996, p. 592). Seealso Hansen and Hodrick (1983) and Frankel (1980).16 Second, table 6 shows that allcountries except Germany experienced low inflation in the 1990s. The volatility ofinflation has also been significantly reduced in New Zealand, Canada, the UK, andSweden. The second term in equation (10), the covariance between the depreciation rateand inflation, is likely to be zero under such circumstances. The last column in table 6reports the correlation between inflation and the depreciation rates, which is clearly verysmall.17

This leaves the first term on the RHS, which is the variance of the depreciation rate,)( 1+∆ tt eVar . Table 6 shows that the volatility measured by the standard deviations is

reduced significantly in all cases except CAD/USD, DM/USD and YEN/USD, where itremained unchanged during the 1990s. Thus, the forward rate unbiasedness hypothesishas a better chance holding in the second sub-samples since inflation rates are low and 16 I actually computed the correlation between real consumption and the exchange rate for New Zealand,

Canada, UK, Sweden and Japan using quarterly data. I then experimented with different values forθ(the constant relative risk-aversion elasticity). I found that these correlations to be small andinsignificant.

17 Also see Obstfeld and Rogoff (1996, p.588 footnote 75) who argue that this is likely to be true because

goods prices )( tP are less variable than asset prices )( te .

stable, the variances of the depreciation rates are small and the risk premia are negligible,particularly in the two inflation-targeting countries, New Zealand and the UK. SouthAfrica experienced a significant reduction in the volatility of the SAR/USD and thecorrelation between its inflation rate and the spot rate were insignificant, but thehypothesis did not hold. I speculate that the risk premium is high in South Africa, whichmakes up the wedge between the forward rate and the expected spot exchange rate. Dataon consumption are not available for South Africa.

There is an argument suggesting that the results of the NZD/USD and the GPB/USD maybe related to the fact that both countries adopted free floating exchange rate policies. Thisargument sounds reasonable, but it cannot explain why the hypothesis fails to hold in theCAD/USD. Canada also adopted a free floating exchange rate policy. Admittedly, moreresearch is needed in this area to explain the mix of results obtained here.

4 Summary

The forward rate unbiasedness hypothesis was tested for New Zealand and other inflation-targeting countries (i.e., Canada, UK and Sweden) and for some non-inflation targeting,but low-inflation countries such as Germany and Japan. South Africa was added to hesample for control purposes. The unbiasedness hypothesis was rejected in almost allcases and in all samples, except for New Zealand and the UK during the period ofinflation targeting.

An implication of the Consumption Asset – Pricing Model (CAPM) was used to explainthe results. Given that risk premium is negligible, the forward rate is equal to theexpected spot rate when the variance of the depreciation rate and the covariance betweenthe depreciation rate and inflation are close to zero. These conditions seem to exist inNew Zealand and the UK. Both countries experienced significant reduction in thevolatility of their spot exchange rates with respect to the U.S. dollar. The correlationbetween the inflation rates and the spot exchange rate were not different from zero.Similar conditions are found in the South African data, but the hypothesis failed to hold. Ispeculate that the risk premium is significant in South Africa.

The problems associated with testing the forward rate unbiasedness hypothesis can beameliorated if one estimates a regression equation with an appropriate dynamicspecification like that found in Naka and Whitney (1995). Unlike Naka and Whitney(1995), the results here were found to be sensitive to the sample period because, amongother factors, changes in the nature of the shocks are likely to affect the variance of thespot rate and the covariance between the spot rate and the price level. Also, theestimation of the non-linear dynamic regression of Naka and Whitney is sensitive tostarting values, which does not ensure that the global optimum are in fact found. Thus,although Naka and Whitney’s elaboration of the dynamics seems appropriate, it does notin fact demonstrate that the unbiasedness hypothesis holds for all countries and allperiods. The premia puzzle remains largely unexplained.

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Table 1: OLS Estimates from e e a b f e ut t t t t− = + − +− − −1 1 1* * *( )

Currency Period NOB *a ,P-value, )0::( *0 =aHWald *b , P-value, )1::( *

0 =bHWald P-value 1,0:: **0 == baHWald DW

NZD/USD 85:10-98:05 152 -0.004 (0.29184) -1.10 (0.0005) (0.0003) 2.09CAD/USD 85:10-98:05 152 -0.001 (0.17527) -1.10 (0.0002) (0.0006) 2.04GBP/USD 85:10-98:05 152 -0.000 (0.85988) -0.16 (0.3732) (0.2167) 1.78SDK/USD 85:10-98:05 152 -0.005 (0.18252) 1.54 (0.5720) (0.3338) 1.53DM/USD 85:10-98:05 152 0.003 (0.31970) 1.04 (0.9646) (0.6036) 1.91YEN/USD 85:10-98:05 152 0.007 (0.03957) -1.84 (0.0069) (0.0212) 1.85SAR/USD 85:10-98:05 152 -0.007 (0.19575) -0.39 (0.1016) (0.2506) 1.72

Sub-Sample 1 *1a ,P-value, )0::( *

0 =aHWald *1b , P-value )1::( *

10 =bHWald P-value 1,0:: *1

*10 == baHWald DW

NZD/USD 85:10-88:12 39 -0.025 (0.18405) -2.83 (0.0220) (0.0075) 2.23CAD/USD 85:10-90:12 63 0.000 (0.69879) -0.79 (0.0633) (0.0017) 2.26GBP/USD 85:10-92:10 84 -0.003 (0.75613) -1.51 (0.2999) (0.1510) 1.83SDK/USD 85:10-92:12 86 0.016 (0.01799) 3.49 (0.0870) (0.0535) 1.40DM/USD 85:10-91:04 67 0.001 (0.83851) 3.71 (0.2359) (0.2160) 1.90YEN/USD 85:10-91:04 67 0.013 (0.08593) -2.80 (0.2165) (0.2265) 1.90SAR/USD 85:10-91:04 67 -0.008 (0.32921) -1.52 (0.1249) (0.2958) 1.75Sub-Sample 2 *

2a ,P-value, )0::( *0 =aHWald *

2b , P-value )1::( *20 =bHWald P-value 1,0:: *

2*20 == baHWald DW

NZD/USD 89:01-98:05 113 0.003 (0.31603) 1.46 (0.5386) (0.5869) 1.71CAD/USD 91:01-98:05 89 -0.002 (0.05253) -0.65 (0.0275) (0.0451) 2.00GBP/USD 93:10-98:05 56 0.002 (0.46896) 0.61 (0.8759) (0.5441) 2.22SDK/USD 93:12-98:05 54 -0.002 (0.57319) -3.16 (0.0201) 0.05743 2.01DM/USD 91:04-98:05 85 -0.001 (0.76207) -0.68 (0.1415) (0.0373) 1.81YEN/USD 91:04-98:05 85 0.004 (0.32761) -1.67 (0.0073) (0.0258) 1.75SAR/USD 91:04-98:05 85 -0.008 (0.27479) -0.17 (0.2460) (0.0579) 1.74

Shaded area interpreted as, the null hypotheses 1* =b could not be rejected at the 5% levels.

Table 2: NLLS Estimates from ttttttt vffbefbaee +−+−−+−=− −−−−− )())(1()1( 21121 ρρPeriod NOB a , (Wald P-values), 0=a b , (Wald P-values), 1=b ρ ,(Wald P-values), 0=ρ Wald P-values , a =0, b =1 DW P Value ARCH12

NZD/USD 85:10-98:05 152 -0.030 (0.03377) 0.92 (0.01280) 0.05 (0.52329) 0.00526 0.3813 22.27*

CAD/USD 85:10-98:05 152 0.002 (0.61759) 1.00 (0.80459) -0.10 (0.90141) 0.59924 0.4421 10.49GBP/USD 85:10-98:05 152 0.046 (0.00406) 0.87 (0.00678) 0.17 (0.06350) 0.00951 0.4390 28.58*

SDK/USD 85:10-98:05 152 -0.120 (0.04617) 0.93 (0.03977) 0.26 (0.00242) 0.06869 0.4822 15.46

DM/USD 85:10-98:05 152 -0.037 (0.00161) 0.92 (0.00049) 0.02 (0.79292) 0.00141 0.4131 2.68YEN/USD 85:10-98:05 152 -0.294 (0.00031) 0.93 (0.00028) 0.07 (0.07566) 0.00112 0.3348 7.79SAR/USD 85:10-98:05 152 -0.008 (0.54508) 0.99 (0.48880) 0.15 (0.05171) 0.73441 0.5597 55.22*

SUB-SAMPLE 11a , (Wald P-values), 01 =a 1b (Wald P-values), 11 =b 1ρ (Wald P-values), 01 =ρ Wald P-values, a1 =0, b1 =1 DW P Value ARCH12

NZD/USD 85:10-88:12 39 -0.048 (0.27493) 0.88 (0.14657) -0.02 (0.91587) 0.03671 0.4042 8.19CAD/USD 85:10-90:12 63 0.004 (0.40967) 0.99 (0.91488) -0.13 (0.30338) 0.00406 0.3411 11.66GBP/USD 85:10-92:10 84 0.065 (0.01518) 0.88 (0.02746) 0.06 (0.63107) 0.02284 0.1632 15.16SDK/USD 85:10-92:12 86 -0.284 (0.01741) 0.84 (0.01510) 0.28 (0.05161) 0.02331 0.1834 28.21*

DM/USD 85:10-91:04 67 -0.482 (0.00871) 0.91 (0.00253) -0.08 (0.52103) 0.00333 0.1796 6.11YEN/USD 85:10-91:04 67 -0.590 (0.00005) 0.88 (0.00004) -0.09 (0.44951) 0.00008 0.2569 8.22SAR/USD 85:10-91:04 67 -0.123 (0.06583) 0.85 (0.05907) 0.23 (0.11903) 0.15048 0.4223 21.00#

SUB-SAMPLE 22a , (Wald P-values), 02 =a 2b , (Wald P-values), 12 =b 2ρ (Wald P-values), 02 =ρ Wald P-values, a2 =0, b2 =1 DW P Value ARCH12

NZD/USD 89:01-98:05 113 -0.013 (0.25968) 0.96 (0.20768) 0.11 (0.20952) 0.32094 0.2495 10.48CAD/USD 91:01-98:05 89 -0.002 (0.64129) 0.99 (0.86232) -0.00 (0.95289) 0.49761 0.4426 10.26GBP/USD 93:10-98:05 56 0.053 (0.07465) 0.88 (0.10000) -0.06 (0.67164) 0.11412 0.4849 14.59SDK/USD 93:12-98:05 54 -0.224 (0.08570) 0.88 (0.08319) 0.15 (0.31219) 0.20082 0.3722 9.64DM/USD 91:05-98:05 85 -0.047 (0.07208) 0.89 (0.06745) 0.15 (0.20295) 0.18668 0.3712 5.26YEN/USD 91:05-98:05 85 -0.330 (0.06296) 0.93 (0.06367) 0.24 (0.03188) 0.17465 0.3313 7.82SAR/USD 91:05-98:05 85 -0.001 (0.92543) 0.99 (0.91036) 0.12 (0.21300) 0.98396 0.4133 11.54

Shaded areas interpreted as, the null hypotheses could not be rejected at the 5% level.Asterisk means significant at the 5% level.# means significant at the 10% level.

Table 3: NLLS Estimates from ttttttt vffbefbaee +−+−−+−=− −−−−− )())(1()1( 21121 ρρCurrency Period NOB a , (Wald P-values), 0=a b , (Wald P-values), 1=b ρ ,(Wald P-values), 0=ρ Wald P-values , a =0, b =1 DW

P ValueARCH12

NZD/DM 85:10-98:05 152 0.003 (0.36297) 0.94 (0.05295) -0.07 (0.36853) 0.12156 0.4266 14.73CAD/DM 85:10-98:05 152 0.014 (0.02993) 0.94 (0.00625) -0.00 (0.91662) 0.01952 0.3972 4.54GBP/DM 85:10-98:05 152 0.044 (0.03810) 0.95 (0.04137) 0.19 (0.01730) 0.11442 0.3339 1.94SDK/DM 85:10-98:05 152 -0.023 (0.22173) 0.98 (0.20397) 0.16 (0.05000) 0.39727 0.5106 1.69YEN/DM 85:10-98:05 152 -0.183 (0.00038) 0.92 (0.00006) -0.70 (0.93227) 0.00018 0.3657 2.07SAR/DM 85:10-98:05 152 -0.314 (0.08617) 0.94 (0.05965) 0.08 (0.32945) 0.14651 0.4207 11.03

SUB-SAMPLE 11a , (Wald P-values), 01 =a 1b (Wald P-values), 11 =b 1ρ (Wald P-values), 01 =ρ Wald P-values, a1 =0, b1 =1 DW

P ValueARCH12

NZD/DM 85:10-88:12 39 -0.025 (0.02958) 0.83 (0.02392) -0.18 (0.27813) 0.06019 0.4154 10.42CAD/DM 85:10-90:12 63 0.046 (0.00650) 0.87 (0.00264) -0.07 (0.58269) 0.00766 0.4084 7.08GBP/DM 85:10-92:10 84 0.123 (0.03602) 0.88 (0.03443) 0.18 (0.16812) 0.10347 0.0929 2.49SDK/DM 85:10-92:12 86 -0.029 (0.57628) 0.97 (0.57770) 0.05 (0.76370) 0.85475 0.0008 2.76YEN/DM 85:10-91:04 67 -0.257 (0.00276) 0.90 (0.00041) -0.12 (0.31999) 0.00037 0.1699 7.23SAR/DM 85:10-91:04 67 -0.066 (0.03123) 0.86 (0.01616) 0.07 (0.62206) 0.03898 0.5277 4.89

SUB-SAMPLE 22a , (Wald P-values), 02 =a 2b , (Wald P-values), 12 =b 2ρ (Wald P-values), 02 =ρ Wald P-values, a2 =0, b2 =1 DW

P ValueARCH12

NZD/DM 89:01-98:05 113 0.000 (0.92173) 0.95 (0.19415) 0.07 (0.48341) 0.40331 0.4530 6.19CAD/DM 91:01-98:05 89 0.013 (0.12552) 0.93 (0.07504) 0.05 (0.67225) 0.20240 0.4075 14.48GBP/DM 93:10-98:05 56 0.017 (0.64524) 0.98 (0.73771) 0.21 (0.15059) 0.40760 0.2508 15.10SDK/DM 93:12-98:05 54 -0.139 (0.15929) 0.90 (0.14694) 0.17 (0.26049) 0.16331 0.2829 2.43YEN/DM 91:05-98:05 85 -0.214 (0.07802) 0.90 (0.07374) 0.15 (0.20075) 0.20195 0.3588 5.46SAR/DM 91:05-98:05 85 -0.008 (0.70797) 0.98 (0.71351) 0.07 (0.51457) 0.93224 0.2704 8.00

Shaded areas interpreted as, the null hypotheses could not be rejected at the 5% levels.

Table 4: Grid Search Estimates ttttttt vffbefbaee +−+−−+−=− −−−−− )())(1()1( 21121 ρρPeriod NOB a , (Wald P-values), 0=a b , (Wald P-values), 1=b ρ ,(Wald P-values), 0=ρ Wald P-values , a =0, b =1

NZD/USD 85:10-98:05 152 -0.52 (0.00) -0.01 (0.00) 0.95 (0.00) 0.00CAD/USD 85:10-98:05 152 -0.30 (0.00) -0.04 (0.00) 0.99 (0.00) 0.00GBP/USD 85:10-98:05 152 0.41 (0.00) 0.17 (0.00) 0.87 (0.00) 0.00SDK/USD 85:10-98:05 152 -0.12 (0.01) 0.94 (0.01) 0.26 (0.02) 0.07

DM/USD 85:10-98:05 152 -0.49 (0.00) 0.02 (0.00) 0.92 (0.00) 0.00YEN/USD 85:10-98:05 152 -4.55 (0.00) 0.05 (0.00) 0.94 (0.00) 0.00SAR/USD 85:10-98:05 152 3.85 (0.00) 0.14 (0.00) 1.00 (0.00) 0.00

SUB-SAMPLE 11a , (Wald P-values), 01 =a 1b (Wald P-values), 11 =b 1ρ (Wald P-values), 01 =ρ Wald P-values, a1 =0, b1 =1

NZD/USD 85:10-88:12 39 -0.53 (0.00) -0.08 (0.00) 0.91 (0.00) 0.00CAD/USD 85:10-90:12 63 0.05 (0.00) -0.16 (0.00) 0.99 (0.00) 0.00GBP/USD 85:10-92:10 84 0.52 (0.00) 0.08 (0.00) 0.91 (0.00) 0.00SDK/USD 85:10-92:12 86 -0.19 (0.01) 0.89 (0.00) 0.07 (0.53) 0.13DM/USD 85:10-91:04 67 -0.05 (0.03) 0.91 (0.01) -0.09 (0.54) 0.09YEN/USD 85:10-91:04 67 -5.37 (0.00) -0.09 (0.00) -0.88 (0.00) 0.00SAR/USD 85:10-91:04 67 -0.12 (0.01) 0.86 (0.01) 0.23 (0.07) 0.02

SUB-SAMPLE 22a , (Wald P-values), 02 =a 2b , (Wald P-values), 12 =b 2ρ (Wald P-values), 02 =ρ Wald P-values, a2 =0, b2 =1

NZD/USD 89:01-98:05 113 -0.01 (0.22) 0.97 (0.16) 0.12 (0.22) 0.40CAD/USD 91:01-98:05 89 -0.36 (0.00) -0.08 (0.00) -0.93 (0.00) 0.00GBP/USD 93:10-98:05 56 0.05 (0.08) 0.89 (0.10) -0.06 (0.67) 0.30SDK/USD 93:12-98:05 54 -1.77 (0.00) 0.11 (0.00) 0.90 (0.00) 0.00DM/USD 91:05-98:05 85 -0.41 (0.00) 0.13 (0.00) 0.91 (0.00) 0.00YEN/USD 91:05-98:05 85 -3.79 (0.00) 0.20 (0.00) 0.94 (0.00) 0.00SAR/USD 91:05-98:05 85 5.29 (0.00) 0.12 (0.00) 1.00 (0.00) 0.00

Shaded areas interpreted as, the null hypotheses could not be rejected at the 5% level.P values are obtained by bootstrapping.

Table 5 Grid Search Estimates ttttttt vffbefbaee +−+−−+−=− −−−−− )())(1()1( 21121 ρρCurrency Period NOB a , (Wald P-values), 0=a b , (Wald P-values), 1=b ρ ,(Wald P-values), 0=ρ Wald P-values , a =0, b =1NZD/DM 85:10-98:05 152 -0.02 (0.00) -0.10 (0.00) 0.92 (0.00) 0.00CAD/DM 85:10-98:05 152 0.23 (0.00) -0.02 (0.00) 0.94 (0.00) 0.00GBP/DM 85:10-98:05 152 0.80 (0.00) 0.19 (0.00) 0.95 (0.00) 0.00SDK/DM 85:10-98:05 152 -1.25 (0.00) 0.15 (0.00) 0.97 (0.00) 0.00YEN/DM 85:10-98:05 152 -0.27 (0.00) 0.94 (0.00) 0.20 (0.02) 0.00SAR/DM 85:10-98:05 152 -0.84 (0.00) 0.18 (0.00) 0.98 (0.00) 0.00

SUB-SAMPLE 11a , (Wald P-values), 01 =a 1b (Wald P-values), 11 =b 1ρ (Wald P-values), 01 =ρ Wald P-values, a1 =0, b1 =1

NZD/DM 85:10-88:12 39 -0.11 (0.00) -0.21 (0.00) 0.83 (0.00) 0.00CAD/DM 85:10-90:12 63 0.05 (0.00) 0.85 (0.000) 0.01 (0.92) 0.01GBP/DM 85:10-92:10 84 0.94 (0.00) 0.13 (0.00) 0.86 (0.00) 0.00SDK/DM 85:10-92:12 86 -1.20 (0.00) 0.06 (0.00) 0.92 (0.00) 0.00YEN/DM 85:10-91:04 67 -3.68 (0.00) 0.16 (0.00) 0.93 (0.00) 0.00SAR/DM 85:10-91:04 67 -0.26 (0.00) 0.28 (0.00) 0.92 (0.00) 0.00

SUB-SAMPLE 22a , (Wald P-values), 02 =a 2b , (Wald P-values), 12 =b 2ρ (Wald P-values), 02 =ρ Wald P-values, a2 =0, b2 =1

NZD/DM 89:01-98:05 113 -0.03 (0.00) 0.07 (0.00) 0.94 (0.00) 0.00CAD/DM 91:01-98:05 89 0.16 (0.00) -0.02 (0.00) 0.91 (0.00) 0.00GBP/DM 93:10-98:05 56 0.93 (0.00) 0.19 (0.00) 0.99 (0.00) 0.00SDK/DM 93:12-98:05 54 -1.29 (0.00) 0.14 (0.00) 0.92 (0.00) 0.00YEN/DM 91:05-98:05 85 -0.35 (0.02) 0.92 (0.02) 0.22 (0.05) 0.08SAR/DM 91:05-98:05 85 -1.05 (0.00) 0.06 (0.00) 0.98 (0.00) 0.00

Shaded areas interpreted as, the null hypotheses could not be rejected at the 5% levels.P values are obtained by bootstrapping.

Table 6: Inflation and the depreciation rateCountry Period NOB Mean STD Student t ( µ

0H ) F (2

0σH ) STD F(

2

0teH ∆σ

)Correlation

),( tt Pe ∆∆New Zealand 85Q1-88Q4 16 2.10 1.04 8.16* 7.90* 9.21 7.82* -0.037

89Q1-98Q1 37 0.65 0.37 3.29 -0.056

Canada 85:10-91:12 63 0.36 0.30 7.21* 2.25* 1.15 0.78 -0.02992:01-98:05 89 0.12 0.20 1.30 0.051

UK 85:10-92:11 84 0.44 0.30 5.84* 2.77* 4.02 4.36* -0.03693:11-98:05 56 0.25 0.18 1.92 -0.010

Sweden 85:10-92:12 86 0.48 0.53 7.15* 5.80* 3.46 1.76* 0.07193:12-98:05 54 0.09 0.22 2.60 0.089

Germany 85:10-91:04 67 0.12 0.19 -3.36# 0.49 3.84 1.55 -0.14491:05-98:05 85 0.23 0.27 3.07 -0.104

Japan 85:10-91:04 67 0.14 0.280 1.69 0.97 4.06 1.64 0.06491:05-98:05 85 0.07 0.283 3.17 0.039

South Africa 85:10-91:04 67 1.18 0.468 7.12* 0.97 4.64 5.86* 0.05791:05-98:05 85 0.73 0.473 1.92 0.156

• The UK departed from the Exchange Rate Mechanism (ERM) in September 1992, and established an inflation-targeting regime in November 1992. For thisreason, I removed 12 months of data from the sample. Sweden floated in November 1992 so 12 months are removed.

• The data are monthly except for New Zealand, which are quarterly.

• Inflation is defined as 100*)( 1−− tt PP , where tP is the natural logarithm of the CPI.

• Depreciation’s rate is defined as 100*)( 1−− tt ee . Exchange rates are in terms of the USD.

• NOB is the number of observations.• STD is the standard deviations.

• Students’ t statistic tests 210 µµ ==H vs. 21: µµ >aH or (#) 21: µµ <aH inflation.

• F tests 22

210 : σσ =H vs. 2

221: σσ >aH .

• Asterisk means significant at the 5% level.

Figure 1: Minimum SSR vs. Grid of Rho ValuesNZD/USD

0.035

0.04

0.045

0.05

0.055

0.06

0.065-0

.3

-0.2

-0.2

-0.1 -0

0.02

0.08

0.15

0.21

0.27

0.34 0.4

0.47

0.53

0.59

0.66

0.72

0.79

0.85

0.91

0.98

1.04

1.11

1.17

Rho

SS

R

Local MinGlobal Min

Figure 2: Minimum SSR vs. Grid of Rho ValuesCAD/USD

0.01

0.012

0.014

0.016

0.018

0.02

0.022

0.024-0

.3

-0.2

-0.2

-0.1 -0

0.02

0.08

0.15

0.21

0.27

0.34 0.

4

0.47

0.53

0.59

0.66

0.72

0.79

0.85

0.91

0.98

1.04

1.11

1.17

Rho

SS

R

Global MinLocal Min

Figure 3: Minimum SSR vs. Grid of Rho ValuesGBP/USD

0.0120.0140.0160.018

0.020.0220.0240.0260.028

-0.3

-0.2

-0.2

-0.1 -0

0.04

0.11

0.17

0.24

0.31

0.38

0.45

0.51

0.58

0.65

0.72

0.79

0.85

0.92

0.99

1.06

1.13

1.19

Rho

SS

R

Local MinGolabl Min

Figure 4: Minimum SSR vs. Grid of Rho ValuesSDK/USD

0.010.015

0.020.025

0.030.035

0.040.045

0.050.055

-0.3

-0.2

-0.2

-0.1 -0

0.06

0.13 0.

2

0.27

0.35

0.42

0.49

0.56

0.63

0.71

0.78

0.85

0.92

0.99

1.07

1.14

Rho

SS

R

Local Min Global Min

Fig

ure 5: M

inim

um

SS

R vs. G

rid o

f Rh

o V

alues

DM

/US

D

0.01

0.03

0.05

0.07

0.09

0.11

0.13

-0.3

-0.2

-0.2

-0.1

-0.1

-0

0.06

0.12

0.18

0.24

0.3

0.36

0.42

0.48

0.54

0.6

0.66

0.72

0.78

0.84

0.9

0.96

1.02

1.08

1.14

1.2

RH

O

SSR

Figure 6: Minimum SSR vs. Grid of Rho ValuesYEN/USD

0.01

0.03

0.05

0.07

0.09

0.11

0.13-0

.3

-0.2

-0.2

-0.1 -0

0.06

0.13 0.

2

0.27

0.35

0.42

0.49

0.56

0.63

0.71

0.78

0.85

0.92

0.99

1.07

1.14

Rho

SS

R

Local Min Global Min

Figure 7: Minimum SSR vs. Grid of Rho ValuesSAR-USD

0.01

0.02

0.03

0.04

0.05

0.06

0.07

-0.3

-0.2

-0.1

-0.1 0

0.08

0.15

0.23

0.31

0.38

0.46

0.53

0.61

0.69

0.76

0.84

0.91

0.99

1.07

1.14

Rho

SS

R

Local MinGlobal Min